To generate the power set (all subsets) of a given set of integers, we can use a recursive function. The power set of a set is the set of all its subsets, including the empty set and the set itself.

Recursive Approach

The recursive approach works by considering each element in the set, and for each element, generating subsets that include the element and those that do not.

Here's the Python code for the recursive function:

pythonCopy code
def generate_power_set(nums):
    # Base case: If the set is empty, the power set contains just the empty set
    if len(nums) == 0:
        return [[]]
    
    # Recursive case: Remove the first element and find power set of the remaining elements
    first_element = nums[0]
    remaining_elements = nums[1:]
    
    # Recursively find the power set of the remaining elements
    subsets_without_first = generate_power_set(remaining_elements)
    
    # Generate subsets that include the first element by adding it to each subset from subsets_without_first
    subsets_with_first = [subset + [first_element] for subset in subsets_without_first]
    
    # The power set is the combination of subsets with and without the first element
    return subsets_without_first + subsets_with_first

# Example usage
nums = [1, 2, 3]
power_set = generate_power_set(nums)
print(power_set)

Explanation

1. Base Case: When the input set is empty (nums is []), the power set is simply a set containing the empty set ([[]]).

2. Recursive Case: For a non-empty input set:

   • The function takes the first element (first_element) of the set.
   • It then recursively computes the power set of the remaining elements (remaining_elements).
   • For every subset in the power set of the remaining elements, two cases are generated: subsets that include the first element (subsets_with_first) and subsets that do not (subsets_without_first).

3. Combining Results: The power set is formed by combining both sets of subsets: those with the first element and those without it.

Example Walkthrough

For nums = [1, 2, 3]:

1. First call: generate_power_set([1, 2, 3])

   • first_element = 1
   • Recursively call generate_power_set([2, 3])

2. Second call: generate_power_set([2, 3])

   • first_element = 2
   • Recursively call generate_power_set([3])

3. Third call: generate_power_set([3])

   • first_element = 3
   • Recursively call generate_power_set([])

4. Base case reached: generate_power_set([]) returns [[]].

5. Now, backtrack and combine subsets:

   • generate_power_set([3]) returns [[], [3]]
   • generate_power_set([2, 3]) returns [[], [3], [2], [2, 3]]
   • generate_power_set([1, 2, 3]) returns [[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]]

Would you like more details or have any questions?

Related Questions:

1. How can this recursive function be modified to handle sets with duplicate elements?
2. How does the time complexity of the recursive solution compare to an iterative approach?
3. Can you write an iterative function to generate the power set?
4. How would you implement the same function in a different programming language, like Java or C++?
5. What are some practical applications of generating power sets in computer science?

Tip:

When dealing with recursion, always consider the base case and how each recursive step simplifies the problem!